Note: for the equations of the ellipse and hyperbola we will use: in the ellipse the "a" term is always greater than the "b" term and c2 = a2 - b2,in the hyperbola the "c" term is always greater or equal to the "a" or "b" term and c2 = a2 + b2.

The "magic number" method is used to solve quadratics that can be factored, i.e., their discriminant b2-4ac is zero.

Extracting roots simply means the "b" term is 0 in ax2+bx+c=0 and you can solve this type of quadratic easiest by getting the squared term by itself and taking the square root of both sides to "extract the rooots". This always works when the linear term (the b term) is missing whether or not the quadratic is factorable. Depending on what is under the square root you may get rational, irrational, or complex roots.